Perturbing free motions on hyper spheres without degeneracy lift

Abstract

We consider quantum motion on S3 perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, l, the ordinary orbital angular momentum value, and another, continuous parameter, b, through which an external scale is introduced. We argue that a loss of the geometric hyper-spherical so(4) symmetry of the free motion occurs that leaves intact the unperturbed hydrogen-like degeneracy patterns characterizing the spectrum under discussion. The argument is based on the observation that the expansions of the Scarf I wave functions for fixed l-values in the basis of properly identified $so(4)$ representation functions are power series in the perturbation parameter, b, in which carrier spaces of dimensionality (K+1)^2 with K varying as K\in [l, N-1], and N being the principal quantum number of the Scarf I potential problem, contribute up to the order O(b^(N-1-K)). Nonetheless, the degeneracy patterns can still be interpreted as a consequence of an effective so(4) symmetry, i.e. a symmetry realized at the level of the dynamic of the system, in so far as from the perspective of the eigenvalue problem, the Scarf I results are equivalently obtained from a Hamiltonian with matrix elements of polynomials in a properly identified so(4) Casimir operator. The scheme applies to any dimension, d.

14 pages. arXiv admin note: substantial text overlap with arXiv:1305.1281

References in corpus (6)

• The Coulomb-Oscillator Relation on n-Dimensional Spheres and Hyperboloids
• Random Walks on Hyperspheres of Arbitrary Dimensions
• The su(1,1) dynamical algebra from the Schrödinger ladder operators for N-dimensional systems: hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator
• Rotational symmetry and degeneracy: a cotangent-perturbed rigid rotator of unperturbed level multiplicity
• Symmetry and degeneracy of the curved Coulomb potential on the S3 ball
• Breaking Pseudo-Rotational Symmetry through ${\mathbf{H}}_{+}^2$ Metric Deformation in the Eckart Potential Problem

Cited by in corpus (2)

• Modelling duality between bound and resonant meson spectra by means of free quantum motions on the de Sitter space time dS4
• Parity violating effects in an exotic perturbation of the rigid rotator